Problem: Simplify; express your answer in exponential form. Assume $p\neq 0, k\neq 0$. $\dfrac{{(pk)^{-1}}}{{(pk^{-3})^{-4}}}$
To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(pk)^{-1} = (p)^{-1}(k)^{-1}}$ On the left, we have ${p}$ to the exponent ${-1}$ . Now ${1 \times -1 = -1}$ , so ${(p)^{-1} = p^{-1}}$ Apply the ideas above to simplify the equation. $\dfrac{{(pk)^{-1}}}{{(pk^{-3})^{-4}}} = \dfrac{{p^{-1}k^{-1}}}{{p^{-4}k^{12}}}$ Break up the equation by variable and simplify. $\dfrac{{p^{-1}k^{-1}}}{{p^{-4}k^{12}}} = \dfrac{{p^{-1}}}{{p^{-4}}} \cdot \dfrac{{k^{-1}}}{{k^{12}}} = p^{{-1} - {(-4)}} \cdot k^{{-1} - {12}} = p^{3}k^{-13}$